Video Transcript
Find the first derivative of the
function π¦ equals negative seven π₯ to the fourth power times the natural log of
six π₯ to the fourth power.
Here, we have the product of two
differentiable functions. The first is negative seven π₯ to
the fourth power and the second is the natural log of six π₯ of the fourth
power. We can, therefore, find the first
derivative of our function by using the product rule. This says that the derivative of
the product of two differentiable functions, π’ and π£, is π’ times dπ£ by dπ₯ plus
π£ times dπ’ by dπ₯. If we let π’ be equal to negative
seven π₯ to the fourth power, then dπ’ by dπ₯ is equal to four times negative seven
π₯ cubed. Thatβs negative 28π₯ cubed. We then let π£ be equal to the
natural log of six π₯ to the fourth power.
So how do we obtain dπ£ by dπ₯? Well, we can do one of two
things. We could spot that we have a
composite function and use the chain rule to find its derivative. Alternatively, we can quote the
general result for the derivative of the logarithm of some function π of π₯. This says that if π¦ is equal to
the natural log of this function π of π₯, then dπ¦ by dπ₯ is equal to π prime of
π₯, the derivative of that function, divided by the original function π of π₯. In this case, π of π₯ is equal to
six π₯ to the fourth power. So π prime of π₯, the derivative
of this, is four times six π₯ cubed, which is 24π₯ cubed. dπ£ by dπ₯ is, therefore,
24π₯ cubed, divided by the original function, six π₯ to the fourth power.
Well, this simplifies quite nicely
to four over π₯. And we can now substitute
everything we have into the formula for the product rule. dπ¦ by dπ₯ is π’ times dπ£ by dπ₯. Thatβs negative seven π₯ to the
fourth power times four over π₯ plus π£ times dπ’ by dπ₯. Thatβs the natural log of six π₯ to
the fourth power times negative 28π₯ cubed. We simplify, and then we factor
negative 28π₯ cubed. And we obtained dπ¦ by dπ₯ to be
negative 28π₯ cubed times the natural logarithm six π₯ to the fourth power plus
one.
